A024356 -id:A024356 - cp's OEIS frontend

A067276 Determinant of n X n matrix containing the first n^2 primes in increasing order.

2, -1, -78, 880, -4656, -14304, -423936, 8342720, 711956736, -615707136, 21057138688, -4663930678272, 211912980656128, -9178450735677440, 40005919124799488, 83013253447139328, -8525111273818357760, -800258888289188708352, -15170733077495639179264

Offset: 1

Rick L. Shepherd, Feb 21 2002

The first column contains the first n primes in increasing order, the second column contains the next n primes in increasing order, etc. Equivalently, first row contains first n primes in increasing order, second row contains next n primes in increasing order, etc. Sequences of determinants of matrices specifically containing primes include A024356 (Hankel matrix), A067549 (first n primes on diagonal, other elements 1), A066933 (cyclic permutations of first n primes in each row) and A067551 (first n primes on diagonal, other elements 0).

			a(3) = -78 because det[[2,7,17],[3,11,19],[5,13,23]] = -78 (= det[[2,3,5],[7,11,13],[17,19,23]], the determinant of the transpose.).

Robert Israel, Table of n, a(n) for n = 1..459

Cf. A024356, A067549, A066933, A067551, A114533.

Cf. A119894, A118770, A118772, A118779.

[ Determinant( Matrix(n, n, [ NthPrime(k): k in [1..n^2] ]) ): n in [1..19] ]; // Klaus Brockhaus, May 12 2010

seq(LinearAlgebra:-Determinant(Matrix(n,n,(i,j) -> ithprime(n*(i-1)+j))),n=1..20); # Robert Israel, Jul 12 2017

Table[ Det[ Partition[ Array[Prime, n^2], n]], {n, 19}] (* Robert G. Wilson v, May 26 2006 *)

for(n=1,20,k=0; m=matrix(n,n,x,y, prime(k=k+1)); print1(matdet(m), ", ")) /* The matrix initialization command above fills columns first: Variables (such as) x and y take on values 1 through n for rows and columns, respectively, with x changing more rapidly and they must be specified even though the 5th argument is not an explicit function of them here. */

from sympy.matrices import Matrix
from sympy import sieve
def a(n):
    sieve.extend_to_no(n**2)
    return Matrix(n, n, sieve[1:n**2+1]).det()
print([a(n) for n in range(1, 20)]) # Indranil Ghosh, Jul 31 2017

A350933 Maximal determinant of an n X n Toeplitz matrix using the first 2*n - 1 prime numbers.

Original entry on oeis.org

1, 2, 19, 1115, 86087, 9603283, 2307021183, 683793949387

Offset: 0

Stefano Spezia, Jan 25 2022

For n X n Hankel matrices the same maximal determinants appear.

			a(2) = 19:
    5    2
    3    5
a(3) = 1115:
   11    2    5
    7   11    2
    3    7   11

Lucas A. Brown, A350932+3.py
Wikipedia, Toeplitz Matrix

Cf. A024356, A318173, A350931, A350932 (minimal), A369946, A369947.

a[n_] := Max[Table[Abs[Det[HankelMatrix[Join[Drop[per = Part[Permutations[Prime[Range[2 n - 1]]], i], n], {Part[per, n]}], Join[{Part[per, n]}, Drop[per, - n]]]]], {i, (2 n - 1) !}]]; Join[{1}, Array[a, 5]] (* Stefano Spezia, Feb 06 2024 *)

a(n) = my(v=[1..2*n-1], m=-oo, d); forperm(v, p, d = abs(matdet(matrix(n, n, i, j, prime(p[i+j-1])))); if (d>m, m = d)); m; \\ Michel Marcus, Feb 08 2024

from itertools import permutations
from sympy import Matrix, prime
def A350933(n): return max(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).det() for p in permutations(prime(i) for i in range(1,2*n))) # Chai Wah Wu, Jan 27 2022

a(5) from Alois P. Heinz, Jan 25 2022

a(6)-a(7) from Lucas A. Brown, Aug 27 2022

A092419 Let k = n-th nonsquare = A000037(n); then a(n) = smallest prime p such that the Kronecker-Jacobi symbol K(k,p) = -1.

Original entry on oeis.org

3, 2, 2, 7, 5, 3, 7, 2, 5, 2, 3, 13, 3, 5, 2, 3, 2, 5, 3, 7, 3, 2, 5, 2, 11, 7, 3, 5, 7, 2, 2, 3, 11, 7, 3, 5, 2, 3, 2, 11, 3, 5, 3, 2, 5, 2, 7, 7, 3, 5, 5, 2, 13, 2, 3, 5, 3, 7, 2, 3, 2, 13, 3, 5, 5, 3, 2, 7, 2, 5, 11, 3, 5, 2, 11, 2, 3, 5, 5, 3, 7, 2, 3, 2, 7, 3, 7, 5, 3, 2, 2, 5, 5, 3, 11, 11, 2, 5, 2, 3, 7

Offset: 1

N. J. A. Sloane, Oct 16 2008

nonn

Maple calls K(k,p) the Legendre symbol.

The old entry with this sequence number was a duplicate of A024356.

H. Cohen, A Course in Computational Number Theory, Springer, 1996 (p. 28 defines the Kronecker-Jacobi symbol).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000037. Records: A067073, A070040. See A144294 for another version.

with(numtheory); f:=proc(n) local M,i,j,k; M:=100000; for i from 1 to M do if legendre(n,ithprime(i)) = -1 then RETURN(ithprime(i)); fi; od; -1; end;

a(n)=my(k=n+(sqrtint(4*n)+1)\2); forprime(p=2,, if(kronecker(k,p)<0, return(p))) \\ Charles R Greathouse IV, Aug 28 2016

Definition corrected Dec 03 2008

A290302 Permanent of Hankel matrix of the first 2n-1 prime numbers.

Original entry on oeis.org

1, 2, 19, 642, 52046, 8012520, 1939769432, 718866359440, 368802888131376, 259389489641608736, 238655237778415792704, 270892331023142039759488, 377446142834845276550334720, 626883335555480085495326908416, 1230454868521341442901679438859264

Offset: 0

Alois P. Heinz, Jul 26 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..22
Wikipedia, Hankel matrix.
Wikipedia, Permanent (mathematics).

Cf. A024356.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> ithprime(i+j-1)))):
seq(a(n), n=0..16);

a[n_]:=Permanent[Table[Prime[i+j-1],{i,n},{j,n}]]; Join[{1}, Array[a, 14]] (* Stefano Spezia, Feb 03 2024 *)

A369946 a(n) is the minimal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

Original entry on oeis.org

1, 2, -19, -1115, -57935, -5696488, -2307021183

Offset: 0

Stefano Spezia, Feb 06 2024

			a(2) = -19:
  2, 5;
  5, 3.
a(3) = -1115:
   3,  7, 11;
   7, 11,  2;
  11,  2,  5.
a(4) = -57935:
   7,  5, 17,  2;
   5, 17,  2,  3;
  17,  2,  3, 11;
   2,  3, 11, 13.

Wikipedia, Hankel matrix.

Cf. A024356, A368351.

Cf. A369947 (maximal), A350933 (maximal absolute value), A369949, A350939 (minimal permanent).

a[n_] := Min[Table[Det[HankelMatrix[Join[Drop[per = Part[Permutations[Prime[Range[2 n - 1]]], i], n], {Part[per, n]}], Join[{Part[per, n]}, Drop[per, - n]]]], {i, (2 n - 1) !}]]; Join[{1}, Array[a, 5]]

a(n) = my(v=[1..2*n-1], m=+oo, d); forperm(v, p, d = matdet(matrix(n, n, i, j, prime(p[i+j-1]))); if (dMichel Marcus, Feb 08 2024

from itertools import permutations
from sympy import primerange, prime, Matrix
def A369946(n): return min(Matrix([p[i:i+n] for i in range(n)]).det() for p in permutations(primerange(prime((n<<1)-1)+1))) if n else 1 # Chai Wah Wu, Feb 12 2024

a(6) from Michel Marcus, Feb 08 2024

A369947 a(n) is the maximal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

Original entry on oeis.org

1, 2, 11, 286, 86087, 9603283, 1764195984

Offset: 0

Stefano Spezia, Feb 06 2024

			a(2) = 11:
  3, 2;
  2, 5.
a(3) = 286:
   3, 11, 5;
  11,  5, 7;
   5,  7, 2.
a(4) = 86087:
   7,  3, 13, 17;
   3, 13, 17,  2;
  13, 17,  2, 11;
  17,  2, 11,  5.

Wikipedia, Hankel matrix.

Cf. A024356, A368352.

Cf. A369946 (minimal), A350933 (maximal absolute value), A369949, A350940 (maximal permanent).

a[n_] := Max[Table[Det[HankelMatrix[Join[Drop[per = Part[Permutations[Prime[Range[2 n - 1]]], i], n], {Part[per, n]}], Join[{Part[per, n]}, Drop[per, - n]]]], {i, (2 n - 1) !}]]; Join[{1}, Array[a, 5]]

a(n) = my(v=[1..2*n-1], m=-oo, d); forperm(v, p, d = matdet(matrix(n, n, i, j, prime(p[i+j-1]))); if (d>m, m = d)); m; \\ Michel Marcus, Feb 08 2024

from itertools import permutations
from sympy import primerange, prime, Matrix
def A369947(n): return max(Matrix([p[i:i+n] for i in range(n)]).det() for p in permutations(primerange(prime((n<<1)-1)+1))) if n else 1 # Chai Wah Wu, Feb 12 2024

a(6) from Michel Marcus, Feb 08 2024

A369949 a(n) is the number of distinct values of the determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

Original entry on oeis.org

1, 1, 3, 59, 2459, 174063, 19141721

Offset: 0

Stefano Spezia, Feb 06 2024

Wikipedia, Hankel matrix.

Cf. A024356, A369946, A369947, A350933.

a[n_] := CountDistinct[Table[Det[HankelMatrix[Join[Drop[per = Part[Permutations[Prime[Range[2 n - 1]]], i], n],{Part[per, n]}], Join[{Part[per, n]}, Drop[per, - n]]]], {i, (2 n - 1) !}]]; Join[{1}, Array[a, 5]]

a(n) = my(v=[1..2*n-1], list=List()); forperm(v, p, listput(list, matdet(matrix(n, n, i, j, prime(p[i+j-1]))));); #Set(list); \\ Michel Marcus, Feb 08 2024

from itertools import permutations
from sympy import primerange, prime, Matrix
def A369949(n): return len({Matrix([p[i:i+n] for i in range(n)]).det() for p in permutations(primerange(prime((n<<1)-1)+1))}) if n else 1 # Chai Wah Wu, Feb 12 2024

a(6) from Michel Marcus, Feb 08 2024

A108078 Determinant of a Hankel matrix with factorial elements.

Original entry on oeis.org

1, 2, 12, 576, 414720, 7166361600, 4334215495680000, 125824009525788672000000, 230121443546659694208614400000000, 33669808475874225917238947767910400000000000, 487707458060712424140716248549520230160793600000000000000

Offset: 0

Paul Max Payton, Jun 03 2005

The term (n=1) is a degenerate case, a matrix with single element 2. This sequence involves products of binomial coefficients and is related to the superfactorial function.

M. J. C. Gover, "The Explicit Inverse of Factorial Hankel Matrices", Department of Mathematics, University of Bradford, 1993

Alois P. Heinz, Table of n, a(n) for n = 0..32
IPJFACT, IPJFACT.
Eric Weisstein's World of Mathematics, Barnes G-Function
Wikipedia, Barnes G-function

Cf. A000178, A002514, A024356, A056886, A056887, A059332, A293707.

% the sequence is easily made by:
for i=1:n det(gallery('ipjfact',i,0)) end
% or, more explicitly, by:
d = 1; for i=1:n-1 d = d*factorial(i+1)*factorial(n-i); end d = d*factorial(n+1);

with(LinearAlgebra):
a:= n-> Determinant(Matrix(n, (i,j)->(i+j)!)):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 05 2015
# second Maple program:
a:= n-> (n+1)! * mul((i+1)!*(n-i)!, i=1..n-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 05 2015

A108078[n_]:=Det[Table[(i+j)!,{i,1,n},{j,1,n}]]; Array[A108078, 20] (* Enrique Pérez Herrero, May 20 2011 *)
Table[BarnesG[n + 1] BarnesG[n + 3], {n, 20}] (* Jan Mangaldan, May 22 2016 *)

a(n) = (n+1)! * Product_{i=1..n-1} (i+1)! * (n-i)!.
a(n) = A059332(n)*(n+1)!.
a(n) ~ n^(n^2 + 2*n + 11/6) * 2^(n+1) * Pi^(n+1) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 16 2016
a(n) = G(n+1) * G(n+3), where G(n) is the Barnes G function. - Jan Mangaldan, May 22 2016

a(0)=1 prepended and some terms corrected by Alois P. Heinz, Dec 05 2015

A350200 Array read by antidiagonals: T(n,k) is the determinant of the Hankel matrix of the 2*n-1 consecutive primes starting at the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 5, -4, -2, 1, 7, 6, 12, 0, 1, 11, -30, -72, 144, 288, 1, 13, 18, 72, 0, 576, -1728, 1, 17, -42, -72, 288, 1152, -7104, -26240, 1, 19, 30, -96, 144, -1248, -11712, 45248, 222272, 1, 23, 22, -188, 488, -112, -11360, 21184, 450432, 1636864

Offset: 0

Pontus von Brömssen, Dec 19 2021

			Array begins:
  n\k|      1      2       3       4       5       6       7       8
  ---+--------------------------------------------------------------
   0 |      1      1       1       1       1       1       1       1
   1 |      2      3       5       7      11      13      17      19
   2 |      1     -4       6     -30      18     -42      30      22
   3 |     -2     12     -72      72     -72     -96    -188    -480
   4 |      0    144       0     288     144     488    1800    2280
   5 |    288    576    1152   -1248    -112    4432   -1552   15952
   6 |  -1728  -7104  -11712  -11360  -10816   29952  -89152  -57088
   7 | -26240  45248   21184 -103168  -43264 -605440 -379264  271552
   8 | 222272 450432 1068800 2022912 3927552 5399552 6315904 6861312
T(3,2) = 12, the determinant of the Hankel matrix
  [3  5  7]
  [5  7 11]
  [7 11 13].

Wikipedia, Hankel matrix

Cf. A350201.

Cf. A000012 (row n = 0), A000040 (row n = 1), A056221 (row n = 2 with opposite sign), A024356 (column k = 1), A071543 (column k = 2).

from sympy import Matrix,prime,nextprime
def A350200(n,k):
    p = [prime(k)] if n > 0 else []
    for i in range(2*n-2): p.append(nextprime(p[-1]))
    return Matrix(n,n,lambda i,j:p[i+j]).det()

Offset corrected by Pontus von Brömssen, Aug 25 2022

cp's OEIS Frontend

A067276 Determinant of n X n matrix containing the first n^2 primes in increasing order.

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A350933 Maximal determinant of an n X n Toeplitz matrix using the first 2*n - 1 prime numbers.

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A092419 Let k = n-th nonsquare = A000037(n); then a(n) = smallest prime p such that the Kronecker-Jacobi symbol K(k,p) = -1.

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A290302 Permanent of Hankel matrix of the first 2n-1 prime numbers.

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A369946 a(n) is the minimal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

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A369947 a(n) is the maximal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

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A369949 a(n) is the number of distinct values of the determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

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